In mathematics, a Priestley space is an orderedtopological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality between the category of Priestley spaces and the category of bounded distributive lattices.
Definition
A Priestley space is an ordered topological space, i.e. a set equipped with a partial order and a topology, satisfying the following two conditions:
Each Priestley space is Hausdorff. Indeed, given two points of a Priestley space, if, then as is a partial order, either or. Assuming, without loss of generality, that, provides a clopen up-set of such that and. Therefore, and are disjoint open subsets of separating and.
Each Priestley space is also zero-dimensional; that is, each open neighborhood of a point of a Priestley space contains a clopen neighborhood of. To see this, one proceeds as follows. For each, either or. By the Priestley separation axiom, there exists a clopen up-set or a clopen down-set containing and missing. The intersection of these clopen neighborhoods of does not meet. Therefore, as is compact, there exists a finite intersection of these clopen neighborhoods of missing. This finite intersection is the desired clopen neighborhood of contained in.
It follows that for each Priestley space, the topological space is a Stone space; that is, it is a compact Hausdorff zero-dimensional space. Some further useful properties of Priestley spaces are listed below. Let be a Priestley space. A Priestley morphism from a Priestley space to another Priestley space is a map which is continuous and order-preserving. Let Pries denote the category of Priestley spaces and Priestley morphisms.
Priestley spaces are closely related to spectral spaces. For a Priestley space, let denote the collection of all open up-sets of. Similarly, let denote the collection of all open down-sets of. Theorem: If is a Priestley space, then both and are spectral spaces. Conversely, given a spectral space, let denote the patch topology on ; that is, the topology generated by the subbasis consisting of compact open subsets of and their complements. Let also denote the specialization order of. Theorem: If is a spectral space, then is a Priestley space. In fact, this correspondence between Priestley spaces and spectral spaces is functorial and yields an isomorphism between Pries and the category Spec of spectral spaces and spectral maps.
Connection with bitopological spaces
Priestley spaces are also closely related to bitopological spaces. Theorem: If is a Priestley space, then is a pairwise Stone space. Conversely, if is a pairwise Stone space, then is a Priestley space, where is the join of and and is the specialization order of. The correspondence between Priestley spaces and pairwise Stone spaces is functorial and yields an isomorphism between the category Pries of Priestley spaces and Priestley morphisms and the category PStone of pairwise Stone spaces and bi-continuous maps. Thus, one has the following isomorphisms of categories: One of the main consequences of the duality theory for distributive lattices is that each of these categories is dually equivalent to the category of bounded distributive lattices.
